Lecture 5. Application of Group Theory to Quantum Mechanics. Part ...

Lecture 5. Application of Group Theory to Quantum Mechanics. Part II 1 Matrix elements of operators 1.1 Irreducible operators

Let us consider a group G of transformations in the space, i.e. for each element G: G 0 ~r ! ~r = G~r :

(1)

These transformations induce the corresponding transformations in the space of wave functions, which we denote by the operator D^ (G): G 0 (~r) ! (~r) = D^ (G) (~r) = (G;1~r) :

(2)

If T^ is an operator in the space of functions, then under transformation of the coordinates G, this operator will also be transformed as G ^0 T^ ! T = D^ (G)T^D^ ;1(G) : (3) Let us suppose that there exists a set of operators T^i() such that each transformed operator will be a superposition of the operators of the set, X G ^ 0() T^i() ! T i = D^ (G)T^i() D^ ;1(G) = Dji() (G)T^j() ; (4) j

where Dji() (G) are the matrix elements of the irreducible representation D() (G). Then the set of operators T^i() is called an irreducible set of operators. The operators T^i() satisfying (4) are said to transform according to the irreducible representation D() (G) of the group G. The number of the operators T^i() is equal to the dimension of the irreducible representation D() (G). In general, if an operator T^ is not irreducible, then it can be represented as a superposition of irreducible operators: X T^ = T^() : (5)

Example 1



Find the transformation properties of the electric dipole operator with respect to the group C3v .

1. The electric dipole operator

d~ = e~r (6) is a vector and therefore is represented by three components, or by a set of three operators: (dx; dy ; dz ). The operators dx = ex, dy = ey, dz = ez transform as operators x, y and z, respectively. It is known that x, y are the basis functions of the irreducible representation E , while z is the basis function of the irreducible representation A1 of the group C3v (see 1

Example 2 of Section 1.4 of lecture 3). This means that two operators (dx; dy ) form an irreducible set and they transform according to the 2-dimensional irreducible representation E , while dz is also an irreducible operator and it transforms according to the 1-dimensional irreducible representation A1 of C3v : dz ! T^(A1 ) (7) dx; dy ! T^1(E); T^2(E) :

Example 2

Find the character of the electric dipole operator with respect to the group SO(2).

Let us consider how the vectors ~ex, ~ey and ~ez transform under operations of the group SO(2). If the axis of rotation coincides with axis z, then D^ (a)~ex = cos a~ex + sin a~ey (8) D^ (a)~ey = ; sin a~ex + cos a~ey ^ D(a)~ez = ~ez or for the linear combinations of ~ex and ~ey we have D^ (a)(~ex + i~ey ) = exp (;ia)(~ex + i~ey ) D^ (a)(~ex ; i~ey ) = exp (ia)(~ex ; i~ey ) D^ (a)~ez = ~ez

(9)

The operator D^ (a) is a rotation operator

@ ): D^ (a) = exp (;a @

(10)

The representations of the group SO(2) can be labelled by the integer m:

Dm(a) = exp (;ima) :

(11)

Therefore, vector ~ez is the basis vector of the irreducible representation D0, while vectors (~ex  i~ey ) are the basis vectors of the irreducible representations D1. This means that operators dx + idy , dx ; idy and dz are irreducible operators and they transform according to the irreducible representations D1, D;1 and D0, respectively, of the group SO(2): dx + idy ! T^1 (12) dx ; idy ! T^;1 0 ^ dz ! T :

Example 3

Find the character of the electric dipole operator with respect to the group SO(3).

Let us de ne the operators

Ylm(~r) = rl Ylm(; ) : 2

(13)

Under rotations, the spherical functions transform as X Ylm(~r) ! D^ (; ; )Ylm(; ) = Dm(l) m(; ; )Ylm (; ) ; m

(14)

0

0

0

i.e. they form a basis of the irreducible representations of the group SO(3). The transformation properties of the operators (13) are similar: X Ylm(~r) ! D^ (; ; )Ylm(~r)D^ ;1(; ; ) = Dm(l) m(; ; )Ylm (~r) : (15) m

0

0

0

This means that operators Ylm(~r) for each l form an irreducible set of operators and they transform according to the irreducible representation D(l) of the group SO(3), or in other words, they are tensors of rank l: Ylm(~r) ! T^(l) : (16) Since q Y10 (~r) = q 43 z (17) Y11 (~r) = q83 (x + iy) 3 Y1;1 (~r) = 8 (x ; iy) three spherical components of the radius-vector x + iy, x ; iy and z form an irreducible set with respect to the group SO(3) and they transform according to the irreducible representation D(1) of this group. Thus, the radius-vector ~r transforms according to the irreducible representation D(1) of SO(3). Since d~ = e~r, the electric dipole operator also transforms according to the irreducible representation D(1) of this group: d~ ! T^(1) : (18)

1.2 Calculation of matrix elements of operators

In order to calculate transition probabilities WIF , we need to calculate the matrix elements of the operator, responsible for this transition between the states of interest, since WIF  jh F jT^j I ij2 : (19)

ALGORITHM

1. Find the symmetry character of initial and nal states, e.g. I = i() and F = k( ) . 2. Find the tensor character of the transition operator, e.g. T^ = T^( ) or decompose it into irreducible components according to (5) if necessary. 3. Calculate the matrix elements of the type h k( ) jT^j( )j i() i : (20) For the matrix elements of the irreducible operators the Wigner-Eckart theorem holds: X h k( ) jT^j( ) j i() i = (i j j kt)h ( )jjT^( )jj () it ; (21) t

where (i j j kt) are the Clebsch-Gordan coecients of the group G and h is called a reduced matrix element. 3

( )

jjT^( )jj

( )

it

CONSEQUENCES OF THE WIGNER-ECKART THEOREM 1. The Wigner-Eckart theorem gives a simple receipt to calculate all matrix elements of the type (20). To do this, we should rst calculate one matrix element (20) for particular values i, j and k. The Clebsch-Gordan coecients for any groups are usually tabulated. So, we can nd a value of the reduced matrix element h ( ) jjT^( )jj () it. Then using the tabulated Clebsch-Gordan coecients, it is simple to compute all the rest matrix elements (20). 2. Often in the calculation of reaction cross-sections or transition probabilities, it is required to know not matrix elements themselves, but the sum of matrix elements on the i or j . Such problems appear in the calculation of reaction cross-sections if the beam and the target are not polarized and the detector is not sensible to the polarization of the incoming particles. The Wigner-Eckart theorem allows to perform the summations automatically.

Example

If the nuclear Hamiltonian is invariant with respect to SO(3) group, then the states can be characterized by a value of the total angular momentum J and they are (2J +1)-fold degenerate. If operator T^(LM ) describes a transition from an initial state ji; JiMi i to a nal state jf ; Jf Mf i, then the reduced transition probability for a given (Ji; Mi) should be averaged over all projections Mi and summed over all values Mf of the nal state and all values M of the transition operator: X B (L; Ji ! Jf ) = 2J 1+ 1 jhf ; Jf Mf jT^(LM )ji ; JiMi ij2 : (22) i Mi ;M;Mf The reduced matrix element of the SO(3) group is de ned with an additional square root in front of it (by some historical reasons), i.e.

hf ; Jf Mf jT^(LM )ji ; JiMi i = (JiMqi JM jJf Mf ) hf ; Jf jjT^(LM )jji ; Jii : 2Jf + 1

(23)

Applying the Wigner-Eckart theorem and using the formulae (21) and (22) for the SO(3) Clebsch-Gordan coecients from the rst lecture, we get X (Ji Mi JM jJf Mf )2 B (L; Ji ! Jf ) = 2J 1+ 1 jhf ; Jf jjT^(L)jji; Jiij2 = 2 J + 1 i f M;Mf 1 jh ; J jjT^(L)jj ; J ij2 : i i 2Ji + 1 f f

(24)

3. Selection rules The matrix element (20) will be zero unless D( ) (G) is contained in the decomposition of D( ) : X D( ) (G) = m D( ) (G) (25)

This means that if T^( ) is a transition operator, then from a state to those states ( ) are allowed for which (25) holds. 4

()

, transitions only

Example 1

Find the selection rules for electric dipole transitions for a system of C3v symmetry.

The states of the system can be associated with the irreducible representations of the group C3v : A1, A2, E . From Example 1 of section 1.1 we know that the components of the electric dipole operator d~ = e~r (26) has the following transformation properties: operators dx and dy transform according to the irreducible representation E , while dz transforms according to the irreducible representation A1 of C3v . From the table

A1  A1 A1 A2  A1 A2 E  A1 E A1  E E A2  E E E  E A1  A2  E we nd the selection rules. The operator of the radiation linear polarized along z axis is dz , transforming as A1 . It can connect only the states which have the same symmetry. So, allowed transitions are

dz : A1 $ A1 ; A2 $ A2 ; E $ E :

(27)

The operator of the radiation polarized in the xy-plane is characterized by dx; dy , transforming as E . The allowed transitions are (dx; dy ) : A1 $ E ; A2 $ E ; E $ E :

(28)

Example 2

Find the selection rules for electric dipole transitions for an axially symmetric system.

The states of the system can be associated with the irreducible representations of the group SO(2) and thus can be labelled by an integer of half-integer m. From Example 2 of section 1.1 we know that the components of the electric dipole operator d~ = e~r (29) has the following transformation properties: operators (dx + idy ), (dx ; idy ) and dz transform according to the irreducible representation D1, D;1 and D0 of SO(2), respectively. For SO(2) the Clebsch-Gordan series (25) looks like follows

Dm1  Dm2 = Dm1 +m2 : 5

(30)

Thus the transitions of linear polarized radiation (dz ) are allowed between the states with
m = 0, while the radiation polarized in the xy-plane (dx  idy ) can be emitted or absorbed only between the states with
m = 1.

Example 3

Find the selection rules for the electric and magnetic multipole transitions in a spherically symmetric system.

The electric and magnetic transition operators have a form: q L (2L+1)!! R ~j (~r)
A ~ ELM (q~r)d~r T^(E ; LM ) = 1c Lq +1 qL R~ T^(M; LM ) = ; ci LL+1 (2Lq+1)!! j (~r)
A~ M L LM (q~r)d~r where q = !=c = 2=, and ~  L~ jL (qr)YLM (~r) A~ ELM (q~r) = qpL1(L+1) r L~ A~ M jL (qr)YLM (~r) : LM (q~r) = p L(L+1)

(31)

(32)

From here it is seen that the operators T^(E ; LM ) and T^(M; LM ) transform according to the irreducible representations D(L) of the group SO(3). From the Clebsch-Gordan series for the SO(3) group

D

(Ji L)

(G) =

JX i +L Jf =jJi;Lj

D(Jf )(G)

(33)

it follows that the transitions from the level Ji will go only to the levels with Jf = jJi ; Lj; : : : ; Ji + L. Suppose that the system is also invariant with respect to inversion I (the group of inversion consists of two elements: identity E and a space inversion I , which corresponds to the transformation x ! ;x; y ! ;y; z ! ;z). The group I has two 1-dimesional irreducible representations,

E I D(1) 1 1 D(2) 1 ;1 If the Hamiltonian of the system is invariant with respect to inversion, then its eigenstates will belong to either the irreducible representation D(1) or to the irreducible representation D(2) . In other words, they will be characterized by a certain parity, +1 or ;1 for P^ (~r) = + (~r) or P^ (~r) = ; (~r) ; (34) respectively, where P^ is a parity operator: P^ (~r) = (;~r) : (35) 6

From the table

D(1)  D(1) D(1) D(2)  D(1) D(2) D(2)  D(2) D(1) the selection rules for a matrix element follows: h f jT^j ii = 0 if PiPT Pf = ;1 :

(36)

From equations (31){(32) follows that T^(E ; LM ) possesses parity (;1)L and T^(M; LM ) possesses parity (;1)L+1 . Thus, the selection rules with respect to the group I are

PiPf = (;1)L for electric 2L pole transitions PiPf = (;1)L+1 for magnetic 2L pole transitions :

(37)

4. Equivalent operator method If operators T^( ) and S^( ) has the same transformation properties with respect to the group G, then their matrix elements are proportional: h k( ) jT^j( ) j i() i = h ( ) jjT^( )jj () i  A ; (38) h k( ) jS^j( )j i() i h ( ) jjS^( )jj () i  where A is just a constant depending on , and . The advantage of the formula (38) is the following. Sometimes it is dicult or even impossible to calculate the matrix elements of an operator T^( ) , but it is simple to calculate the matrix elements of an equivalent operator S^( ) . Then, the matrix elements of interest are h k( ) jT^j( )j i() i = A h k( ) jS^j( ) j i() i : (39)

Example

Calculate the spin-orbit splitting in a many-electron atom.

The Hamiltonian of a many-electron atom can be presented as

H=

X

i

!

2 X Ze2 X e2 X h  ; 2m
i ; r + r + f (ri)(~li
~si) ; i i i = X hj1 j2; j jj[U (k1 )  V (k2 )](k) jj0j10 j20 ; j 0i = >: j10 j20 j 0 >;   k1 k2 k q ( k ) 00 0 00 (2j + 1)(2j 0 + 1)(2k + 1)hj jjU 1 jj j ih j jjV (k2 ) jj0j 0 i

(60)

00

1

2

1

2

(the sum over 00 is included in case both operators acts on the same quantum number ). 4. For the scalar product of two operators U (k) and V (k) acting on the coordinates of the functions related to the angular momenta j1 and j2 , respectively, we have

hj1j2 ; jmj(U
V (k)

(k )

X



00

)j0j 0 j 0 ; j 0m0i = 

jj mm

1 2

0

(;1)j2 +j1+2k+j 0

0

hj1jjU (k) jj00j10 ih00j2 jjV (k) jj0j20 i :

10

(

)

j1 j2 j  j20 j10 k (61)

3 Conservation laws The observable T is said to conserve if its mean value does not change in time in any state (~r; t). d d ^ ^ ^d dt h jT j i = h dt jT j i + h jT j dt i = i1h f;hH^ jT^j i + h jT^jH^ ig (62) 1 ^ H^ ]j i = 0 : = ih h j[T; This means that the observable T conserves if the operator T^ commutes with the Hamiltonian. If the system possesses a certain symmetry, i.e. the Hamiltonian is invariant with respect to a group G, then for all elements G, the transformation operators D(G) commute with the Hamiltonian. At rst glance, this suggests an existence of a huge number of conserved quantities. However, not all of them are independent. For example, the conservation of D(G3) relates to the conservation of D(G1) and D(G2) if G1
G2 = G3. So, if the group G is of order n and m  n elements of the group generate the rest of n elements, then the system will have only m conserved values, or integrals of motion.

Example

All elements of the group C3 can be generated from one element C3, since C3
C3 = C32, C3
C32 = E . Thus, the invariance of a system with respect to the group C3 will give rise to one conserving quantity.

References [1] L.D.Landau, E.M.Lifshitz, Quantum Mechanics. Non-Relaticistic Theory. (AddisonWesley Publishing Company, Inc., reading, MA., 1958). [2] E.Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra (Academic, New York, 1959). [3] M.Hamermesh, Group Theory and its Application to Physical Problems (AddisonWesley Reading, MA, 1962). [4] J.P.Elliott, P.G.Dawber, Symmetry in Physics (The Macmillan Press, London, 1979). [5] A.Frank, P. Van Isacker, Algebraic Methods in Molecular and Nuclear Structure Physics (John Wesley and Sons, Inc, 1994).

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